practice

1. Write something.
2. Write something else..
3. The modified photographs below show a counterweighted, steel drawbridge in the closed (span down) and open (span up) positions. The elements used to solve this problem are highlighted in color: pivot points in red, lever arms in yellow, and forces in green.

   Bridge Closed -- Span Down [magnify]		Bridge Open -- Span Up [magnify]

   • The red triangles represent the pivot points for …
     • the counterweight assembly on the left and
     • the bridge span on the right.
   • The yellow lines highlight the lever arms.
     • The line running along the roadbed is divided into intervals of equal length, L.
     • The other lines have lengths that can be determined with geometric reasoning if we assume that the angles between beams are all multiples of 45° (that is, 0°, 45°, 90°, 135°, 180°).
   • The green arrows show the relevant forces.
     • W_b and W_c are the weights of the moveable bridge span and counterweight, respectively.
     • T₁ and T₂ are the tensions in the linkage when the bridge is closed and open, respectively.
     • When the bridge is closed the moveable span is balanced so that there is no normal force on the far end.
   Determine the following quantities in terms of the weight of the bridge span.
   1. The weight of the counterweight and the tension when the bridge is closed.
   2. The tension when the bridge is open and the torque needed to keep the span open.
4. Write something completely different.

conceptual

1. How can all three types of equilibrium -- stable, unstable, and neutral -- be demonstrated using …
   1. an egg
   2. a cone
   3. a torus (a shape like a donut or a bagel)
2. Explain the reasoning behind each of the following general rules of design.
   1. Aircraft carriers are designed to be stable in the ocean.
   2. Fighter planes are designed to be unstable in flight.

statistical

1. center-of-population.txt
   The center of population of the United States as defined by the Census Bureau is the same as that of the center of gravity of a collection of point masses on a plane. It is the point at which a weightless, smooth, spherical shell in the shape of the "lower 48" states and the District of Columbia would balance if weights of identical size were placed on it -- each weight representing the location of one person. On such an imaginary surface, north-south distances between parallels of latitude (ϕ) are identical and their angular measure in degrees may be used as units of displacement. In contrast, east-west distances between meridians of longitude (λ) are not constant but vary with latitude from a maximum at the equator to zero at the poles. Multiplying by the cosine of the latitude will correct for this convergence of the meridians at the poles. In addition, small areas of the country are used as data points rather than individual human beings, which reduces the computational burden. (In 1960 43,000 areas were used but by 2000, this number had risen to more than 8,000,000. By 2020 or 2030, the number of areas will probably equal the number of residents.)
   
   Thus, the center of population of the US computed by the Census Bureau is the point whose latitude (φ) and longitude (λ) satisfy the equations …

   φ =	∑ wiφi		λ =	∑ wiλicos φi
   	∑ wi			∑ wicos φi

   Where ϕ_i, λ_i, and w_i are the latitude, longitude, and population of the census areas included in the calculation.
   1. The data on the accompanying tabs-delimited text file give the population and the effective latitude and longitude in degrees of the fifty states and the District of Columbia from the 2000 census. Using this data, determine the coordinates of the population center of the United States at this time. (Be sure to exclude Alaska and Hawaii from your calculations, but do include the District of Columbia.)
   2. In what state is this point located? Which county? What is the nearest incorporated community (city, village, or town)? What is the nearest street intersection?
   3. Go to this location and await further instructions.

investigative

1. Determine the mass of a ruler using a known weight to balance it over a pivot.

worksheets

1. The Physics Teacher has published several articles containing free body diagram worksheets. They are available free to members of the American Association of Physics Teachers (AAPT). Everyone else has to pay.
   1. Free-body diagrams revisited — II. James E. Court. The Physics Teacher. Vol. 37, No. 8 (November 1999): 490-495. Note: pages 490-493 are relevant to this topic.